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arXiv:1407.7572 · Physical Review B

Shear viscosity and imperfect fluidity in bosonic and fermionic superfluids

Rufus Boyack, Hao Guo, Kathryn Levin

Published Dec 1, 2014• 2 citations
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Abstract

In this paper we address the ratio of the shear viscosity to entropy density $\ensuremath{\eta}/s$ in bosonic and fermionic superfluids. A small $\ensuremath{\eta}/s$ is associated with nearly perfect fluidity, and more general measures of the fluidity perfection/imperfection are of wide interest to a number of communities. We use a Kubo approach to concretely address this ratio via low-temperature transport associated with the quasiparticles. Our analysis for bosonic superfluids utilizes the framework of the one-loop Bogoliubov approximation, whereas for fermionic superfluids we apply BCS theory and its BCS-BEC extension. Interestingly, we find that the transport properties of strict BCS and Bogoliubov superfluids have very similar structures, albeit with different quasiparticle dispersion relations. While there is a dramatic contrast between the power law and exponential temperature dependence for $\ensuremath{\eta}$ alone, the ratio $\ensuremath{\eta}/s$ for both systems is more similar. Specifically, we find the same linear dependence (on the ratio of temperature $T$ to inverse lifetime $\ensuremath{\gamma}(T))$ with $\ensuremath{\eta}/s\ensuremath{\propto}T/\ensuremath{\gamma}(T)$, corresponding to imperfect fluidity. By contrast, near the unitary limit of BCS-BEC superfluids a very different behavior results, which is more consistent with near-perfect fluidity.

DOI
10.1103/physrevb.90.214501

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In this paper we address the ratio of the shear viscosity to entropy density \ensuremathη/s\ensuremath{\eta}/s\ensuremathη/s in bosonic and fermionic superfluids. A small \ensuremathη/s\ensuremath{\eta}/s\ensuremathη/s is associated with nearly perfect fluidity, and more general measures of the fluidity perfection/imperfection are of wide interest to a number of communities. We use a Kubo approach to concretely address this ratio via low-temperature transport associated with the quasiparticles. Our analysis for bosonic superfluids utilizes the framework of the one-loop Bogoliubov approximation, whereas for fermionic superfluids we apply BCS theory and its BCS-BEC extension. Interestingly, we find that the transport properties of strict BCS and Bogoliubov superfluids have very similar structures, albeit with different quasiparticle dispersion relations. While there is a dramatic contrast between the power law and exponential temperature dependence for \ensuremathη\ensuremath{\eta}\ensuremathη alone, the ratio \ensuremathη/s\ensuremath{\eta}/s\ensuremathη/s for both systems is more similar. Specifically, we find the same linear dependence (on the ratio of temperature TTT to inverse lifetime \ensuremathγ(T))\ensuremath{\gamma}(T))\ensuremathγ(T)) with \ensuremathη/s\ensuremath∝T/\ensuremathγ(T)\ensuremath{\eta}/s\ensuremath{\propto}T/\ensuremath{\gamma}(T)\ensuremathη/s\ensuremath∝T/\ensuremathγ(T), corresponding to imperfect fluidity. By contrast, near the unitary limit of BCS-BEC superfluids a very different behavior results, which is more consistent with near-perfect fluidity.
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We compute the shear viscosity of the unitary Fermi gas above the superfluid transition temperature, using a diagrammatic technique that starts from the exact Kubo formula. The formalism obeys a Ward identity associated with scale invariance which guarantees that the bulk viscosity vanishes identically. For the shear viscosity, vertex corrections and the associated Aslamazov-Larkin contributions are shown to be crucial to reproduce the full Boltzmann equation result in the high-temperature, low fugacity limit. The frequency dependent shear viscosity η(ω)\eta(\omega)η(ω) exhibits a Drude-like transport peak and a power-law tail at large frequencies which is proportional to the Tan contact. The weight in the transport peak is given by the equilibrium pressure, in agreement with a sum rule due to Taylor and Randeria. Near the superfluid transition the peak width is of the order of 0.5TF0.5 T_F0.5TF​, thus invalidating a quasiparticle description. The ratio η/s\eta/sη/s between the static shear viscosity and the entropy density exhibits a minimum near the superfluid transition temperature whose value is larger than the string theory bound ℏ/(4πkB)\hbar/(4\pi k_B)ℏ/(4πkB​) by a factor of about seven.
Shear viscosity is a measure of the amount of dissipation in a simple fluid. In kinetic theory shear viscosity is related to the rate of momentum transport by quasi-particles, and the uncertainty relation suggests that the ratio of shear viscosity η to entropy density s in units of ℏ/ k B is bounded by a constant. Here, ℏ is Planck's constant and k B is Boltzmann's constant. A specific bound has been proposed on the basis of string theory where, for a large class of theories, one can show that η/ s ⩾ ℏ/(4π k B ). We will refer to a fluid that saturates the string theory bound as a perfect fluid. In this review we summarize theoretical and experimental information on the properties of the three main classes of quantum fluids that are known to have values of η/ s that are smaller than ℏ/ k B . These fluids are strongly coupled Bose fluids, in particular liquid helium, strongly correlated ultracold Fermi gases and the quark gluon plasma. We discuss the main theoretical approaches to transport properties of these fluids: kinetic theory, numerical simulations based on linear response theory and holographic dualities. We also summarize the experimental situation, in particular with regard to the observation of hydrodynamic behavior in ultracold Fermi gases and the quark gluon plasma.
We review the modern view of fluid dynamics as an effective low-energy, long-wavelength theory of many-body systems at finite temperature. We introduce the concept of a nearly perfect fluid, defined by a ratio η/s of shear viscosity to entropy density of order ℏ/kBor less. Nearly perfect fluids exhibit hydrodynamic behavior at all distances down to the microscopic length scale of the fluid. We summarize arguments that suggest that there is fundamental limit to fluidity, and we review the current experimental situation of measurements of η/s in strongly coupled quantum fluids.
Perfect fluids are characterized as having the smallest ratio of shear viscosity to entropy density, $\ensuremath{\eta}/s$, consistent with quantum uncertainty and causality. So far, nearly perfect fluids have only been observed in the quark-gluon plasma and in unitary atomic Fermi gases, exotic systems that are amongst the hottest and coldest objects in the known universe, respectively. We use angle resolved photoemission spectroscopy to measure the temperature dependence of an electronic analog of $\ensuremath{\eta}/s$ in an optimally doped cuprate high-temperature superconductor, finding it too is a nearly perfect fluid around, and above, its superconducting transition temperature Tc{T}_{c}Tc​.